Topics in string amplitudes Harold Erbin Universit di Torino (Italy) - - PowerPoint PPT Presentation

Topics in string amplitudes

Harold Erbin

Università di Torino (Italy)

In collaboration with: – Corinne de Lacroix – Juan Maldacena – Ashoke Sen – Dimitri Skliros arXiv: 1810.07197, 1906.06051

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Outline: 1. Introduction

Introduction Two-point amplitude Crossing symmetry: QFT Crossing symmetry: string theory Conclusion

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Properties of string theory

String theory = theory of extended objects ◮ consistency? (unitarity, crossing symmetry. . . ) ◮ differences with local point-particle QFT? ◮ non-locality?

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Properties of string theory

String theory = theory of extended objects ◮ consistency? (unitarity, crossing symmetry. . . ) ◮ differences with local point-particle QFT? ◮ non-locality? Point-particle QFT ◮ consistency assessed from S-matrix ◮ locality ∼ analyticity of S-matrix

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Properties of string theory

String theory = theory of extended objects ◮ consistency? (unitarity, crossing symmetry. . . ) ◮ differences with local point-particle QFT? ◮ non-locality? Point-particle QFT ◮ consistency assessed from S-matrix ◮ locality ∼ analyticity of S-matrix

String theory properties

• 1. if possible, direct proof
• 2. otherwise, prove property consequence → indirect test

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Properties of string theory

String theory = theory of extended objects ◮ consistency? (unitarity, crossing symmetry. . . ) ◮ differences with local point-particle QFT? ◮ non-locality? Point-particle QFT ◮ consistency assessed from S-matrix ◮ locality ∼ analyticity of S-matrix

String theory properties

• 1. if possible, direct proof
• 2. otherwise, prove property consequence → indirect test

Natural framework: string field theory (off-shell, renormalization. . . )

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Plan

Properties of (super)string amplitudes:

• 1. Tree-level 2-point amplitude

with: Juan Maldacena, Dimitri Skliros [1906.06051]

• 2. Analyticity and crossing symmetry at all loops

with: Corinne de Lacroix, Ashoke Sen [1810.07197]

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2-point amplitude

◮ QFT A2(k, k′) = 2k0(2π)D−1 δ(D−1)(k − k′) (1-particle state normalization, cluster decomposition)

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2-point amplitude

◮ QFT A2(k, k′) = 2k0(2π)D−1 δ(D−1)(k − k′) (1-particle state normalization, cluster decomposition) ◮ string theory A2 ∼ 1 Vol SL(2, C)

• d2zd2z′ Vk(z, ¯

z)Vk′(z′, ¯ z′)

• S2

∼ 1 Vol R+ Vk(∞, ∞)Vk′(0, 0)S2

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2-point amplitude

◮ QFT A2(k, k′) = 2k0(2π)D−1 δ(D−1)(k − k′) (1-particle state normalization, cluster decomposition) ◮ string theory (standard lore) A2 ∼ 1 Vol SL(2, C)

• d2zd2z′ Vk(z, ¯

z)Vk′(z′, ¯ z′)

• S2

∼ 1 Vol R+ Vk(∞, ∞)Vk′(0, 0)S2 = 0 BRST point of view: need Ngh = 6 but only 2 operators c¯ cV

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2-point amplitude

◮ QFT A2(k, k′) = 2k0(2π)D−1 δ(D−1)(k − k′) (1-particle state normalization, cluster decomposition) ◮ string theory (standard lore) A2 ∼ 1 Vol SL(2, C)

• d2zd2z′ Vk(z, ¯

z)Vk′(z′, ¯ z′)

• S2

∼ 1 Vol R+ Vk(∞, ∞)Vk′(0, 0)S2 = 0 BRST point of view: need Ngh = 6 but only 2 operators c¯ cV QFT result is universal → how to resolve contradiction?

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2-point amplitude

◮ QFT A2(k, k′) = 2k0(2π)D−1 δ(D−1)(k − k′) (1-particle state normalization, cluster decomposition) ◮ string theory (standard lore) A2 ∼ 1 Vol SL(2, C)

• d2zd2z′ Vk(z, ¯

z)Vk′(z′, ¯ z′)

• S2

∼ 1 Vol R+ Vk(∞, ∞)Vk′(0, 0)S2 = 0 BRST point of view: need Ngh = 6 but only 2 operators c¯ cV QFT result is universal → how to resolve contradiction? Vk(∞, ∞)Vk′(0, 0)S2 ∝ δ(0) δ(D−1)(k − k′) = ∞ from on-shell + momentum conservation → ambiguous, need regularization / better gauge fixing

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Analyticity and crossing symmetry

Analyticity of n-point amplitude An(k1, . . . , kn) ◮ starting point for other properties (crossing symmetry, dispersion relations) ◮ related to locality and causality

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Analyticity and crossing symmetry

Analyticity of n-point amplitude An(k1, . . . , kn) ◮ starting point for other properties (crossing symmetry, dispersion relations) ◮ related to locality and causality Crossing symmetry: ◮ relations between amplitudes with exchange of particles/anti-particles in initial/final states ◮ often assumed or observed (scattering amplitude program. . . )

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Analyticity and crossing symmetry

Analyticity of n-point amplitude An(k1, . . . , kn) ◮ starting point for other properties (crossing symmetry, dispersion relations) ◮ related to locality and causality Crossing symmetry: ◮ relations between amplitudes with exchange of particles/anti-particles in initial/final states ◮ often assumed or observed (scattering amplitude program. . . ) Why a general proof? ◮ ensure observed examples not accident of simple amplitudes ◮ learn about fundamental properties of QFT

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Method

Proof idea in QFT [Bros-Epstein-Glaser, ’64-65]:

• 1. prove analyticity of S-matrix in “primitive domain” ∆
• 2. analytic extension H(∆)
• 3. show that 2) ⇒ crossing symmetry

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Method

Proof idea in QFT [Bros-Epstein-Glaser, ’64-65]:

• 1. prove analyticity of S-matrix in “primitive domain” ∆
• 2. analytic extension H(∆)
• 3. show that 2) ⇒ crossing symmetry

Remarks: ◮ 1) is non-perturbative (full S-matrix) ◮ 2) and 3) are general statements from theory of several complex variables

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Method

Proof idea in QFT [Bros-Epstein-Glaser, ’64-65]:

• 1. prove analyticity of S-matrix in “primitive domain” ∆
• 2. analytic extension H(∆)
• 3. show that 2) ⇒ crossing symmetry

Remarks: ◮ 1) is non-perturbative (full S-matrix) ◮ 2) and 3) are general statements from theory of several complex variables String theory: ◮ interactions non-locality → no position space Green functions ◮ prove 1) perturbatively from Feynman diagrams

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Outline: 2. Two-point amplitude

Introduction Two-point amplitude Crossing symmetry: QFT Crossing symmetry: string theory Conclusion

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Gauge-fixed amplitude

◮ 2-point amplitude A0,2(k, k′) = 8πα′−1 Vol K0

• d2zd2z′ Vk(z, ¯

z)Vk′(z′, ¯ z′)

• S2

K0 := PSL(2, C)

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Gauge-fixed amplitude

◮ 2-point amplitude A0,2(k, k′) = 8πα′−1 Vol K0

• d2zd2z′ Vk(z, ¯

z)Vk′(z′, ¯ z′)

• S2

K0 := PSL(2, C) ◮ simple gauge-fixing A0,2(k, k′) = 8πα′−1 Vol K2 Vk(∞, ∞)Vk′(0, 0)S2 K2 := U(1) × R+ = dilatation × rotation

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Gauge-fixed amplitude

◮ 2-point amplitude A0,2(k, k′) = 8πα′−1 Vol K0

• d2zd2z′ Vk(z, ¯

z)Vk′(z′, ¯ z′)

• S2

K0 := PSL(2, C) ◮ simple gauge-fixing A0,2(k, k′) = 8πα′−1 Vol K2 Vk(∞, ∞)Vk′(0, 0)S2 K2 := U(1) × R+ = dilatation × rotation ◮ evaluate CFT correlation function + regularize zero-modes A2(k, k′) = lim

κ0→0(2π)D−1δ(D−1)(k + k′) 16π2i δ(κ0)

α′ Vol K2

Normalization: Vk(z, ¯ z)Vk′(z′, ¯ z′)S2 = i (2π)Dδ(D)(k + k′) |z − z′|4 . numerator = zero-modes ei(k+k′)·x for Lorentzian target spacetime

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Compute CKV volume (1)

◮ Volume regularization Vol K2 =

d2z

|z|2 = 2

2π

dθ

∞

dr r = 4π

∞

−∞

dτ = 4π lim

ε→0

∞

−∞

dτ eiετ Volε K2 = 8π2 δ(ε) ◮ (τ, ε) Euclidean worldsheet (time, energy) on the cylinder (dimensionless) ◮ problem: Lorentzian spacetime, dimensionful energy → need Wick rotation and rescaling

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Compute CKV volume (2)

Jacobian from mode expansions without oscillators:

• 1. worldsheet Wick rotation

τ = it, ε = −iE

• 2. Lorentzian regularized volume

VolM,E K2 = 8π2i δ(E)

• 3. Lorentzian mode expansion

X 0 = x0 + α′k0t

• 4. scale between spacetime and worldsheet times / energies

t = ξ0 α′k0 = ⇒ E = α′k0κ0 (ξ0, κ0) dimensionful worldsheet variables

• 5. regularized Lorentzian volume

VolM,κ0 K2 = 8π2i δ(κ0) α′k0

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Result

A2(k, k′) = lim

κ0→0(2π)D−1δ(D−1)(k + k′) 16π2i δ(κ0)

α′ VolM,κ0 K2 VolM,κ0 K2 = 8π2i δ(κ0) α′k0 Recover QFT result: A2(k, k′) = 2k0(2π)D−1δ(D−1)(k + k′)

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Result

A2(k, k′) = lim

κ0→0(2π)D−1δ(D−1)(k + k′) 16π2i δ(κ0)

α′ VolM,κ0 K2 VolM,κ0 K2 = 8π2i δ(κ0) α′k0 Recover QFT result: A2(k, k′) = 2k0(2π)D−1δ(D−1)(k + k′) Remarks: ◮ regularization ambiguous → fixed from unitarity ◮ Jacobian can be computed from path integral (field shift) ◮ better approach: gauge fix X 0 [1906.06051] ◮ operator approach [1909.03672, Seki-Takahashi] ◮ can always insert 6 ghosts example: using 0| c−1¯ c−1c0¯ c0c1¯ c1 |0 = 1 A2(k, k′) = CS2 Vol K2 c¯ cVk(∞, ∞)c0¯ c0 c¯ cVk′(0, 0)S2

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Zero-point amplitude

Next step

Generalization to 0-point function → compute on-shell action ◮ Zero-point amplitude for Minkowski spacetime M: A0[M] ∼ δ(D)(0) Vol SL(2, C)

?

= ∞

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Zero-point amplitude

Next step

Generalization to 0-point function → compute on-shell action ◮ Zero-point amplitude for Minkowski spacetime M: A0[M] ∼ δ(D)(0) Vol SL(2, C)

?

= ∞ ◮ curved background X: e−(SEH[X]−SEH[M]) = A0[X] A0[M]

?

= finite (à la Gibbons–Hawking–York) ◮ consider X = black hole?

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Outline: 3. Crossing symmetry: QFT

Introduction Two-point amplitude Crossing symmetry: QFT Crossing symmetry: string theory Conclusion

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Amplitude and Green functions

4-point scattering process ◮ pa = (Ea, pa) ∈ C, a = 1, . . . , 4: external momenta ◮ momentum conservation: p1 + · · · + p4 = 0 ◮ on-shell condition: p2

a = −m2 a

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Amplitude and Green functions

4-point scattering process ◮ pa = (Ea, pa) ∈ C, a = 1, . . . , 4: external momenta ◮ momentum conservation: p1 + · · · + p4 = 0 ◮ on-shell condition: p2

a = −m2 a

Green functions:

• ff-shell

G(p1, . . . , p4) = truncated ˜ G(p1, . . . , p4) = G(p1, . . . , p4)

4

• a=1

(p2

a + m2 a)

• n-shell

A(p1, . . . , p4) = lim

p2

a→−m2 a

˜ G(p1, . . . , p4) QFT: G = sum of Feynman diagrams

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Physical amplitudes

Mandelstam variables s = −(p1 + p2)2, t = −(p1 + p3)2, u = −(p1 + p4)2 mass-shell: s + t + u =

a m2 a

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Physical amplitudes

Mandelstam variables s = −(p1 + p2)2, t = −(p1 + p3)2, u = −(p1 + p4)2 mass-shell: s + t + u =

a m2 a

Physical regions ◮ S (s-channel): s ≥

a m2 a,

t, u ≤ 0 ◮ T (t-channel): t ≥

a m2 a,

s, u ≤ 0 ◮ U (u-channel): u ≥

a m2 a,

s, t ≤ 0 Physical amplitudes AS,T,U(p1, . . . , p4) = lim

pa∈S,T,U A(p1, . . . , p4)

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Mandelstam plane

pa ∈ R on-shell

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Statement of crossing symmetry

Crossing symmetry

The processes S : 1 + 2 → 3 + 4 T : 1 + ¯ 3 → ¯ 2 + 4 U : 1 + ¯ 4 → 3 + ¯ 2 (and CPT conjugates) are equivalent under analytic continuation on the complex mass-shell AS(s, t) = AT(t, s), AS(s, u) = AU(u, s)

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Statement of crossing symmetry

Crossing symmetry

The processes S : 1 + 2 → 3 + 4 T : 1 + ¯ 3 → ¯ 2 + 4 U : 1 + ¯ 4 → 3 + ¯ 2 (and CPT conjugates) are equivalent under analytic continuation on the complex mass-shell AS(s, t) = AT(t, s), AS(s, u) = AU(u, s) ◮ looks natural from LSZ: AS,T,U all come from a single function A ◮ but: not guaranteed that A is analytic in a domain with paths between S, T, U

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QFT proof (1)

Outline of proof [Bros-Epstein-Glaser ’64-65][Bros ’86]:

• 1. assumptions: m2

a > 0, asymptotic states = stable particles

• 2. define the “primitive domains”

∆k =

• Aα

Im P(α) = 0, (Im P(α))2 ≤ 0

• ∪

Im P(α) = 0, −P2

(α) < M2 α

• ∩

Im pi

a = 0, i = k, . . . , D − 1

• Aα ⊂ {1, . . . , n},

P(α) =

a∈Aα

pa, Mα: production threshold

In words: pa with k possible complex components s.t. all Pα have: 1) non-zero imaginary timelike part, or 2) real momentum squared below multi-particle threshold in channel Aα

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QFT proof (2)

• 3. prove analyticity inside ∆D of S-matrix from micro-causality

(fields commute at spacelike separations) [Araki, Burgoyne,

Ruelle, Steimann, ’60-61]

problem: ∆D ∩ mass-shell = ∅

• 4. compute the “envelope of holomorphy” H(∆2) (= analytic

extension) → H(∆2) ∩ mass-shell = ∅

• 5. show ∃ a path in H(∆2) ∩ mass-shell between all pairs of

iǫ-neighbourhoods of physical regions

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QFT proof (2)

• 3. prove analyticity inside ∆D of S-matrix from micro-causality

(fields commute at spacelike separations) [Araki, Burgoyne,

Ruelle, Steimann, ’60-61]

problem: ∆D ∩ mass-shell = ∅

• 4. compute the “envelope of holomorphy” H(∆2) (= analytic

extension) → H(∆2) ∩ mass-shell = ∅

• 5. show ∃ a path in H(∆2) ∩ mass-shell between all pairs of

iǫ-neighbourhoods of physical regions Notes: ◮ only H(∆2) is necessary ◮ 4) and 5) ⇐ theory of several complex variables only ◮ work with the complete S-matrix

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Primitive domain and mass-shell

Proof that ∆D ∩ mass-shell = ∅ :

• 1. complex mass-shell:

Re pa · Im pa = 0, (Re pa)2 − (Im pa)2 + m2

a = 0

• 2. if Im pa timelike, (Im pa)2 ≤ 0, then need Re pa timelike,

(Re pa)2 < 0, for 2nd condition, but violates 1st condition

• 3. if Im pa = 0, then −P2

(α) ≥ M2 α

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Envelope of holomorphy

More on the envelope of holomorphy: ◮ consider f (z1, . . . , zn) analytic in ∆ ◮ analyticity in several variables ⇒ constrain shape of ∆ ◮ if shape not arbitrary: analyticity in ∆ ⇒ analyticity in H(∆) ◮ given ∆, H(∆) is independent of f ◮ typically: use edge-of-the-wedge theorem (Bogoliubov)

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Outline: 4. Crossing symmetry: string theory

Introduction Two-point amplitude Crossing symmetry: QFT Crossing symmetry: string theory Conclusion

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String field theory

◮ field theory (second-quantization) ◮ rigorous, constructive formulation [hep-th/9206084, Zwiebach] ◮ make gauge invariance explicit (L∞ algebras et al.) ◮ use standard QFT techniques (renormalization, analyticity. . . ) → prove consistency (Cutkosky rules, unitarity, soft theorems, background independence. . . ) [Sen ’14-19] ◮ help to compute worldsheet scattering amplitudes [Sen ’14-19] and effective actions [1912.05463, HE-Maccaferri-Vošmera] ◮ study backgrounds (= classical solutions), marginal and RR fluxes deformations [1811.00032, Cho-Collier-Yin; 1902.00263, Sen] ◮ access collective, non-perturbative, thermal, dynamical effects ◮ worldvolume theory ill-defined for (p > 1)-branes

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String field theory

◮ field theory (second-quantization) ◮ rigorous, constructive formulation [hep-th/9206084, Zwiebach] ◮ make gauge invariance explicit (L∞ algebras et al.) ◮ use standard QFT techniques (renormalization, analyticity. . . ) → prove consistency (Cutkosky rules, unitarity, soft theorems, background independence. . . ) [Sen ’14-19] ◮ help to compute worldsheet scattering amplitudes [Sen ’14-19] and effective actions [1912.05463, HE-Maccaferri-Vošmera] ◮ study backgrounds (= classical solutions), marginal and RR fluxes deformations [1811.00032, Cho-Collier-Yin; 1902.00263, Sen] ◮ access collective, non-perturbative, thermal, dynamical effects ◮ worldvolume theory ill-defined for (p > 1)-branes

To appear: “String Field Theory – A Modern Introduction”, Lecture Notes in Physics, Springer

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SFT in a nutshell

SFT = standard QFT s.t.: ◮ infinite number of fields (of all spins) ◮ infinite number of interactions ◮ non-local interactions ∝ e−#k2 ◮ reproduce worldsheet amplitudes (if well-defined)

[1703.06410, De Lacroix-HE-Kashyap-Sen-Verma]

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SFT in a nutshell

SFT = standard QFT s.t.: ◮ infinite number of fields (of all spins) ◮ infinite number of interactions ◮ non-local interactions ∝ e−#k2 ◮ reproduce worldsheet amplitudes (if well-defined)

[1703.06410, De Lacroix-HE-Kashyap-Sen-Verma]

Consequences of non-locality: ◮ cannot use position representation ◮ cannot use assumptions from local QFT (micro-causality. . . ) ◮ cannot derive analyticity like in QFT

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SFT in a nutshell

SFT = standard QFT s.t.: ◮ infinite number of fields (of all spins) ◮ infinite number of interactions ◮ non-local interactions ∝ e−#k2 ◮ reproduce worldsheet amplitudes (if well-defined)

[1703.06410, De Lacroix-HE-Kashyap-Sen-Verma]

Consequences of non-locality: ◮ cannot use position representation ◮ cannot use assumptions from local QFT (micro-causality. . . ) ◮ cannot derive analyticity like in QFT → study Green function singularities from Feynman diagrams in momentum space

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Action and Feynman diagrams

◮ gauge-fixed action S = 1 2 Ψ| c−

0 c+ 0 L+ 0 |Ψ +

• g,n≥0

gg2g−2+n

s

n! Vg,n(Ψn)

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Action and Feynman diagrams

◮ gauge-fixed action S = 1 2 Ψ| c−

0 c+ 0 L+ 0 |Ψ +

• g,n≥0

gg2g−2+n

s

n! Vg,n(Ψn) ◮ propagator A1| b+ L+ b−

0 |A2

= ◮ fundamental g-loop n-point vertex Vg,n(A1, . . . , An) = defined s.t. sum of all graphs ⇒ recover worldsheet amplitudes

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Example

?

= + +

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Example

= + + +

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Momentum representation (1)

◮ string field Fourier expansion |Ψ =

• A
• dDk

(2π)D φA(k) |A, k k: D-dimensional momentum A: discrete labels (Lorentz indices, group repr., KK modes. . . ) ◮ 1PI action S =

• dDk φA(k)KAB(k)φB(−k)

+

• n
• dDk1 · · · dDkn V (n)

A1,...,An(k1, . . . , kn)φA1(k1) · · · φAn(kn)

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Momentum representation (2)

Propagator KAB(k)−1 = −i MAB k2 + m2

A

QA(k) ◮ MAB mixing matrix for states of equal mass ◮ QA polynomial

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Momentum representation (3)

Vertices − iV (n)

A1,...,An(k1, . . . , kn) = −i

• dt e

−g{Aa}

ij

(t) ki·kj−c

n

• a=1

m2

a

× PA1,...,An

k1, . . . , kn; t

• ◮ t moduli parameters

◮ P{Aa} polynomial ◮ c > 0 → damping in sum

• ver states

◮ gij positive definite ◮ no singularity for ki ∈ C (finite) ◮ lim

k0→±i∞ V (n) = 0

◮ lim

k0→±∞ V (n) = ∞

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Green function

Truncated Green function = sum of Feynman diagrams of the form F(p1, . . . , pn) ∼

• dT
• s

dDℓs e−Grs(T) ℓr·ℓs−2Hra(T) ℓr·pa−Fab(T) pa·pb ×

• i

1 k2

i + m2 i

P(pa, ℓr; T) T, moduli parameters, P, polynomial in (pa, ℓr) ◮ momenta:

◮ external {pa} ◮ internal {ki} ◮ loop {ℓs}

ki = linear combination of {pa, ℓs} ◮ Grs positive definite

◮ integrations over spatial loop momenta ℓr converge ◮ integrations over loop energies ℓ0

r diverge

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Momentum integration

Prescription = generalized Wick rotation [1604.01783, Pius-Sen]:

• 1. define Green function for Euclidean internal/external momenta
• 2. analytic continuation of external energies + integration

contour s.t.

◮ keep poles on the same side ◮ keep ends at ±i∞

→ analyticity for pa ∈ R, p0

a in first quadrant Im p0 a > 0, Re p0 a ≥ 0

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Momentum integration

Prescription = generalized Wick rotation [1604.01783, Pius-Sen]:

• 1. define Green function for Euclidean internal/external momenta
• 2. analytic continuation of external energies + integration

contour s.t.

◮ keep poles on the same side ◮ keep ends at ±i∞

→ analyticity for pa ∈ R, p0

a in first quadrant Im p0 a > 0, Re p0 a ≥ 0

− →

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Momentum integration

Prescription = generalized Wick rotation [1604.01783, Pius-Sen]:

• 1. define Green function for Euclidean internal/external momenta
• 2. analytic continuation of external energies + integration

contour s.t.

◮ keep poles on the same side ◮ keep ends at ±i∞

→ analyticity for pa ∈ R, p0

a in first quadrant Im p0 a > 0, Re p0 a ≥ 0

− → ⇒ Cutkosky rules, unitarity, spacetime and moduli space iǫ-prescriptions [Pius, Sen] Timelike Liouville theory [1905.12689, Bautista-Dabholkar-HE]

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Analyticity for string theory (1)

Result

Analyticity inside ∆2 of n-point superstring Green functions at all loop orders: ◮ implies crossing symmetry for n = 4 ◮ identical analyticity properties for QFT and string theory

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Analyticity for string theory (1)

Result

Analyticity inside ∆2 of n-point superstring Green functions at all loop orders: ◮ implies crossing symmetry for n = 4 ◮ identical analyticity properties for QFT and string theory Comments: ◮ Feynman graphs → perturbative computations ◮ valid for states with any spin ◮ technical assumptions: mass gap, stable external states ◮ regularization of massless states: removes IR non-analyticity (identical to QFT)

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Analyticity for string theory (2)

Method to study singularity:

• 1. start with some pa = p(1)

a , ℓ0 r ∈ iR, ℓr ∈ R s.t. no singularity

• 2. find a path pa = p(1)

a

→ desired pa = p(2)

a

• 3. deform the integral contour as the poles move
• 4. assume ∃ singularity = on-shell internal propagator

pinching = collision of two poles from opposite sides

• 5. analyze reduced diagram, display an inconsistency

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Analyticity for string theory (2)

Method to study singularity:

• 1. start with some pa = p(1)

a , ℓ0 r ∈ iR, ℓr ∈ R s.t. no singularity

• 2. find a path pa = p(1)

a

→ desired pa = p(2)

a

• 3. deform the integral contour as the poles move
• 4. assume ∃ singularity = on-shell internal propagator

pinching = collision of two poles from opposite sides

• 5. analyze reduced diagram, display an inconsistency

Proceed by steps:

• 1. analyticity in ∆1: go from pa = 0 to desired Re pa and Im p0

a

(keep Im pa = 0)

• 2. analyticity in ∆2: go from pa ∈ ∆1 to desired Im p1

a (keep

Im pi

a = 0 ∀i ≥ 2)

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First step

◮ p0

a ∈ C, pa ∈ R

◮ pinching implies reduced graph: k2

i = −m2 i , arrow = sign of k0 i

◮ pa, ℓr ∈ R ⇒ ki ∈ R, then k2

i = −m2 i ⇒ ki ∈ R

◮ one can prove ∀i : k0

i > 0

◮ implies P(α) =

• i

ki ∈ R, p2

a = −m2 a

⇒ −P2

(α) ≥ M2 α

→ contradiction – one must have −P2

(α) < M2 α

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Second step

◮ p

a = (p0 a, p1 a) ∈ C, p⊥ a ∈ R

◮ pinching implies reduced graph: arrow = sign of Im k1

i

◮ one can prove ∀i : Im k1

i > 0, and

k2

i = −m2 i

⇒ Im k

i ∈ W + ⇒ Im P(α) =

• i

Im ki ∈ W + → contradiction – one must have Im P(α) timelike

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Outline: 5. Conclusion

Introduction Two-point amplitude Crossing symmetry: QFT Crossing symmetry: string theory Conclusion

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Conclusion

Results: ◮ tree-level 2-point amplitude computation consistent with QFT ◮ analyticity of superstring n-point amplitudes in ∆2 ◮ proof of crossing symmetry for 4-point superstring amplitudes at the same level as in QFT ◮ show that, in some sense, string theory behaves like local QFT ◮ new proof of analyticity valid for more general QFTs

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Conclusion

Results: ◮ tree-level 2-point amplitude computation consistent with QFT ◮ analyticity of superstring n-point amplitudes in ∆2 ◮ proof of crossing symmetry for 4-point superstring amplitudes at the same level as in QFT ◮ show that, in some sense, string theory behaves like local QFT ◮ new proof of analyticity valid for more general QFTs Outlook: ◮ tree-level 0-point function for generic background ◮ CPT theorem ◮ analyticity in ∆D ◮ analyticity in H(∆2) just from Feynman diagrams

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